What is the integral of (1-u^2)/(u+u^3) ?

The integral of (1-u^2)/(u+u^3) is ln|u|-ln|u^2+1|+C

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integral of (1-u^2)/(u+u^3)

\int \frac{1 - u ^{2}}{u + u ^{3}} d u

ln ∣ u ∣ - ln u^{2} + 1 + C

Solution steps

\int \frac{1 - u ^{2}}{u + u ^{3}} d u

Take the partial fraction of

\frac{1 - u ^{2}}{u + u ^{3}} : \frac{1}{u} - \frac{2 u}{u ^{2} + 1}

= \int \frac{1}{u} - \frac{2 u}{u ^{2} + 1} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{u} d u - \int \frac{2 u}{u ^{2} + 1} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{2 u}{u ^{2} + 1} d u = ln u^{2} + 1

= ln ∣ u ∣ - ln u^{2} + 1

Add a constant to the solution

= ln ∣ u ∣ - ln u^{2} + 1 + C

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What is the integral of (1-u^2)/(u+u^3) ?
The integral of (1-u^2)/(u+u^3) is ln|u|-ln|u^2+1|+C